Plasma and Fusion Research: Regular Articles

Volume 11, 2401048 (2016)

Reduction Rate of the MHD Potential Energy by a Toroidal Plasma
Current Drive during Tokamak Plasma Formation with DCCS∗)
Osamu WATANABE
Research Institute for Applied Mechanics, Kyushu University, 6-1 Kasuga-koen, Kasuga-shi, Fukuoka 816-8580, Japan(Received 30 October 2015 / Accepted 11 March 2016)
The reduction rate of the magnetohydrodynamic (MHD) potential energy δWM is defined as the energy
diﬀerence between the vacuum magnetic field energy before a tokamak plasma formation and the MHD potential
energy of an equilibrium tokamak plasma. Stationary direct current in a central solenoidal coil (DCCS ) can reduceδWM , which means a reduction of the required heat energy for the tokamak plasma formation. The increase of
the hole component ration in a toroidal plasma current distribution increases the toroidal plasma current and thestored thermal energy without increase of the plasma size. For tokamak plasmas with DCCS = 100 A, δWM has alocal minimum with respect to the toroidal plasma current distribution. In this state, a perturbation of the toroidal
plasma current is energetically inhibited.
c 2016 The Japan Society of Plasma Science and Nuclear Fusion ResearchKeywords: DCCS , expulsion of magnetic field energy, reduction rate of MHD potential energy
DOI: 10.1585/pfr.11.2401048

1. Introduction
A temporally changing current is applied to the central solenoidal coil (CS coil) in the central axis of a usual
tokamak, generating a temporally changing magnetic fieldenergy in order to drive a toroidal plasma current for Joule
heating in induction heating experiments. On the otherhand, no current is applied to the CS coil for non-induction
heating experiments because a tokamak plasma formationand its sustainment by non-induction heating have been
investigated for steady state operation at a constant equilibrium state and for the elimination of the CS coil [1–6].
The CS coil current has little influence on the equilibrium
configuration of tokamak plasmas because a solenoidal
coil ideally only generates a small leakage magnetic field.
However, the magnetohydrodynamic (MHD) potential energy of the equilibrium tokamak plasma in non-induction
heating experiment also depends on the CS coil current.
The CS coil current is an important factor for thestability of the tokamak plasma even if the current is astationary direct current that has been applied before thestart of heating [7]. Similar to the MHD potential energy,
the transition energy of the tokamak plasma formation depends on both the CS current and the time evolution of thetoroidal plasma current. During the temporal change ofthe toroidal plasma current by non-induction heating, themagnetic field energy in the inner region of the CS coil(CS region) changes and influences the plasma in the outerregion of the CS coil through the temporal change of its
vector potential; this process is similar to Joule heating.
Since a stationary direct current in the CS coil (DCCS ) canauthor’s e-mail: rwata@riam.kyushu-u.ac.jp∗) This article is based on the presentation at the 25th International Toki
Conference (ITC25).

enhance the vertical magnetic field only only in the CS region, a reverse of the vertical magnetic field in the CS region at the tokamak plasma formation can be changed by
DCCS to a decrease of the vertical magnetic field [7].
Based on the diﬀerence of the transition energies of
the tokamak plasma formation, the eﬀect of the CS current on the tokamak plasma formation by non-induction
heating is investigated. The reduction rate of the MHD
potential energy δWM is defined as the energy diﬀerence
between the vacuum magnetic field energy before the tokamak plasma formation WM0 and the MHD potential energyof an equilibrium tokamak plasma WM . δWM is normalized to WM0 . The reduction rates are compared based on
the distributions of the toroidal plasma current and DCCS .
In this paper, section 2 and 3 describe the calculation
method and calculation results, respectively. Conclusions
and discussions are presented in section 4. A summary isgiven in section 5.

2. Calculation Method
The force balance in the direction of the magnetic fieldat an equilibrium configuration for a steady state tokamak
plasma is derived from the ideal MHD equations; it depends on the plasma pressure. A toroidal plasma current
has no driving term except the pressure gradient driving
term. The functional form of the plasma pressure P ineq. (1) has two components; Gaussian (n1 , σ1 ) and hole(n2 , σ2 ) components of the toroidal plasma current distribution.
 1Ψ − Ψax
P (Ψ ) = 2T 0 n1 exp
σ1 Ψax − Ψsep2401048-1

c 2016 The Japan Society of Plasma
Science and Nuclear Fusion Research

Plasma and Fusion Research: Regular Articles⎧2 ⎫⎤⎪⎪⎪
Ψ − Ψax ⎪⎨ 1⎬⎥⎥⎥⎥+n2 exp ⎪−⎥.
⎪⎪⎩ σ2 Ψax − Ψsep ⎪⎭⎦Volume 11, 2401048 (2016)(1)

Here, n1 and n2 are plasma densities. σ1 = 0.28 andσ2 = 0.48 define plasma pressures at a mid-plane of an
inboard limiter. T 0 is the plasma temperature. Ψ is themagnetic flux surface function, and Ψax and Ψsep are themagnetic flux surface functions at the magnetic axis andX-points, respectively. The magnetic flux surface function
is selected, after repeated equilibrium calculations in orderto minimize the MHD potential energy WM :P
1 2+dVB ≡ WTH + WB , (2)WM =γ − 1 2μ0
where WTH and WB are the stored thermal energy and thetotal magnetic field energy of the equilibrium tokamak
plasma, respectively. μ0 and γ = 5/3 are the vacuum permeability and the ratio of the specific heats, respectively.
The reduction rate δWM of the MHD potential energy
is defined as follows.
B2
WM −WM0×100, WM0 = dV 0 . (3)δWM =WM0
2μ0Here, B0 is the vacuum magnetic field without the tokamak plasma. δWM is normalized to the total of the vacuum
magnetic field energy WM0 , which is the energy state before the tokamak plasma formation. The same size plasma
with diﬀerent toroidal plasma current has diﬀerent WM0 ,because the poloidal field (PF) coils current is varied toconverge to the local minimum energy states while sustaining the specified positions [8]. WM0 depends on not only
the plasma size but also the plasma pressure profile ration1 /(n1 + n2 ).
A coil configuration of TST2 [5,6] is utilized; the configuration comprises a CS coil with 239 coil turns and threepairs of PF coils with 9 (PF1), 6 (PF2), and 40 (PF3) coils
turns in one-side which are numbered beginning at the topof the device, as shown in Figs. 1 (A-1) and (C-1). Since
positions and currents of the coils are symmetric with respect to the mid-plane, a magnetic axis must exist at Z = 0.
In addition, the position of the two X-points is the same except for the sign of the height Z. Therefore, the specified
positions have three degrees of freedom that can be independently turned by the adjustment of the currents applied
to the three PF coils.
For all calculations, a plasma temperature of T 0 =100 eV and a density of n1 +n2 = 3×1016 m−3 are used. Theradii of the inboard and outer limiters are 0.13 and 0.63 m,respectively. The height of the plasma region is limited toz = ± 0.5 m. It is assumed that there is no plasma outside
the region of the limiters and the last closed flux surface(LCFS).

3. Calculation Result
In the relaxation method, all tokamak plasma equilibria in this study converge to a local minimum of the MHD

Fig. 1 Comparison of equilibrium tokamak plasma shapes. Figure series (A-) and (C-) show situations without DCCS
and with DCCS = −100 A, respectively. Magnetic flux
surfaces and last closed flux surfaces are indicated bysolid and dashed lines in Figs. 1 (A-1) and (C-1), respectively. Triangle and cross marks indicate the magnetic
axes and X-points, respectively. The PF and CS coilblocks are indicated by squares. The plasma pressuresand toroidal plasma current distributions are indicated bysolid and dashed lines in Figs. 1 (A-2) and (C-2). In Figures 1 (A-3) and (C-3), the vertical magnetic fields BV
before and after the tokamak plasma formations are indicated by dashed and solid lines, respectively. Horizontal
axes are in radial direction and are common for all figures.

potential energy by adjusting a plasma pressure profile ateach toroidal plasma current distribution ratio n2 /(n1 + n2 ).
In parallel, the current applied to the three PF coils is adjusted in order to sustain a specified magnetic axis andtwo X-points which determine the shape of the LCFS. In
the calculations of the tokamak plasma equilibrium, in order to sustain the specified positions corresponding to thethree degrees of freedom, the three currents applied to thePF coils are adjusted, as indicated by the dashed lines inFigs. 2 (A,B,C-4).
Both dependencies of the hole current profile component n2 on the stored thermal energy WTH and on
the toroidal plasma current for the equilibrium tokamak
plasma are independent of DCCS . Judging from theLCFSs indicated by the chained lines in Figs. 1 (A-1) with(DCCS = 0 A, n2 /(n1 + n2 ) = 0) and (C-1) with (DCCS =100 A, n2 /(n1 + n2 ) = 0.9), the volumes of the tokamak
plasma are almost the same, even if both toroidal plasma
current distributions and DCCS are diﬀerent. Therefore,2401048-2

Plasma and Fusion Research: Regular Articles

Volume 11, 2401048 (2016)

Fig. 2 Dependencies of energies and currents on the ratio of Gaussian n1 to hole n2 components of the toroidal plasma current distribution.
Solid and other lines correspond to the left and right of the vertical axes, respectively. The magnification ratios of the verticalaxes of the respective left and right parts are the same for (A- ), (B- ), and (C- ), except for Fig. 2(C-1). The (A- ), (B- ), and(C- ) series correspond to DCCS = 0, −50 and −100 A cases, respectively. The reduction rate of the MHD potential energy
δWM is indicated by a combination of circle marks and solid lines. The diﬀerence of the magnetic field energy WB − WM0 is
indicated by dashed lines in Figs. 2 (A,B,C-1). The MHD potential energies WM and their stored thermal energies WTH are plottedin Figs. 2 (A,B,C-2). Figures 2 (A,B,C-3) indicate the magnetic field energies after (WB ) and before (WM0 ) the tokamak plasma
formation. In Figures 2 (A,B,C-4), the toroidal plasma current hardly depends on DCCS , and the PF coils currents PF1, PF2 andPF3 are indicated by dashed, dotted and short dashed lines, respectively. Here, the current of each PF coil is the product of the coilcurrent and the coil turns.

WTH indicated by the dashed line in Figs. 2 (A,B,C-2) andthe toroidal plasma current indicated by the solid line inFigs. 2 (A,B,C-4) hardly depend on the DCCS .
A comparison of Figs. 1 (A-2) and (C-2) shows thatn2 expands the plasma pressure profile around the magnetic axis indicated by solid lines. Therefore, WTH and thetoroidal plasma current are also increased by n2 , as shownin Figs. 2 (A,B,C-2,4). To sustain the increased WTH , thevacuum magnetic field energy WM0 generated by the coils
current is increased as the dashed lines in Figs. 2 (A,B,C3). As a result, the total magnetic field energy including
the toroidal plasma current WB and MHD potential energy
WM = WB + WTH are increased by n2 .
To reduce the MHD potential energy WM , the tokamak
plasma has two magnetic energy expulsion mechanisms
due to the toroidal plasma current drive. Figures 1 (A,C3) show that the vacuum vertical magnetic field outside(inside) the region of the magnetic axis indicated by thedashed lines increases (reverses), and changes into thesolid line profile due to the toroidal plasma current driveat the tokamak plasma formation. In Fig. 1 (C-3), the vacuum vertical magnetic field in the CS region is enhanced
by DCCS = −100 A and is reduced by the toroidal plasma
current drive. Therefore, in all n2 cases, δWM is reduced by
DCCS as shown in Fig. 3. In addition, the toroidal plasma
current expels the vertical magnetic field not only in theCS region but also around the magnetic axis, as shown inFig. 1 (C-3).
δWM with DCCS = −100 A has a local minimum with

Fig. 3 The dependence of the reduction rate on the DCCS . (B)and (C) are extended figures of (A) at DCCS = 0 and −100
A, respectively. δWM for current distributions n2 /(n1 +n2 ) = 0, 0.5 and 1 are indicated by solid, dashed anddotted lines, respectively.

respect to the toroidal plasma current at n2 /(n1 + n2 ) = 0.4,
as shown in Fig. 2 (C-1). On the other hand, δWM withDCCS = 0 and −50 A decreases and increases with increasing n2 , respectively, as shown in Figs. 2(A,B-1). δWMwith DCCS = −50 A at the center of the horizontal axisin Fig. 3 (A) is indicated in Figs. 2 (B-1). Similarly, δWM
with DCCS = 0 and −100 A at the left and right limits ofthe horizontal axis in Figs. 3 (B) and (C) are indicated inFigs. 2 (A-1) and (C-1), respectively. Since δWM is normalized to DCCS , the eﬀect of the magnetic field energy

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Volume 11, 2401048 (2016)

decrease in the CS region is relatively decreased, as shownin Fig. 3.

4. Conclusions and Discussions
The reduction rate of the MHD potential energy δWM
is defined by normalizing the energy necessary for equilibrium tokamak plasma formation WM − WM0 to the vacuum magnetic field energy WM0 for the tokamak plasmaconfinement, as expressed by eq. 3. The tokamak plasma
equilibrium converges to a local minimum of the MHD potential energy WM by applying the relaxation method forthe magnetic flux surface function Ψ , as defined in eq. 1.
The plasma pressure profile P in eq. 1 is given by a Gaussian n1 and hole n2 current component. Simultaneously,
the magnetic axis and two X-points are sustained together
by the adjustment of the three PF coil currents in order toform a uniform volume of an equilibrium tokamak plasma.
The PF1 coil current is also modified to cancel the leakage
magnetic field of the DCCS .
DCCS reduces δWM , as shown in Fig. 3. Since an expulsion of the magnetic field energy by the toroidal plasma
current is amplified by DCCS , as shown in Fig. 1 (C-3),
DCCS hardly influences the equilibrium of the tokamak
plasma. This is because ideally there is no leakage of magnetic field due to the CS coil at the plasma region. Therefore, without modification of the tokamak plasma equilibrium, DCCS reduces the required external heat energy forthe tokamak plasma formation.
An increase of the hole current component n2 increases the capacity for the storing thermal energy WTH in agiven tokamak plasma volume, as shown in Figs. 2 (A,B,C2), and expands the expulsion area of the magnetic fieldenergy around the magnetic axis, as shown in Fig. 1 (C-3).
Therefore, in the case of DCCS = 0 A, δWM decreases as n2increases, as shown in Fig. 2 (A-1). Since the vertical magnetic field in the CS region is reversed, the magnetic fieldenergy decrease in the CS region is small. The magnetic
field energy eﬃciency of the tokamak plasma confinement
is improved increasing n2 , but the equilibrium tokamak
plasma can energetically return to the vacuum state solely
by consuming WTH .
In the case of DCCS = −50 A case, the required energy
reduction rate is negative below the toroidal plasma currentof 0.9 kA at n2 /(n1 + n2 ) = 0.6, as shown in Figs. 2 (B-1,4).
Below the toroidal plasma current, an external energy injection is necessary to return to the vacuum state from thetokamak plasma equilibrium state. Conversely, the equilibrium tokamak plasma can return to the vacuum state by
only consuming the stored thermal energy, if the toroidal
plasma current exceeds 0.9 kA.
Although the volumes of the equilibria tokamak plasmas in the calculations are made uniform by the specification of the magnetic axis and two X-points, the toroidal
plasma current increases with increase in the hole current
component n2 due to the rise of the stored thermal energy

WTH . For the case n2 /(n1 +n2 ) = 0.4 with DCCS = −100 A,
as in Fig. 2 (C-1), a perturbation of the toroidal plasma current is accompanied by an increase of WM /WM0 becauseδWM ∝ WM /WM0 − 1. If the toroidal plasma current increases due to the perturbation, the MHD potential energy
WM also increases. Therefore, the toroidal plasma current
can return to n2 /(n1 + n2 ) = 0.4 state by rejecting the extra
perturbation energy. In contrast, when the toroidal plasma
current decreases due to the perturbation, WM /WM0 increases, even if all coils currents are constant. In that case,
WM -WM0 in Fig. 2 (C-1) further increases with decreasing
toroidal plasma current. As a result, at n2 /(n1 + n2 ) = 0.4,
the toroidal plasma current can be energetically recoveredfrom both an increase and a decrease of toroidal plasma
current.
During an experiment, if all coil currents are constant,a tokamak plasma can absorb thermal energy with not onlya shift of plasma positions but also modifications of theplasma pressure profile. The shift of plasma positions
means a shift of the magnetic axis and the X-points witha change of the plasma volume. In any plasma volume,
DCCS can improve the stability of the tokamak plasma
equilibrium by preparing a local minimum state of δWM ,which depends on the non-induction heating power.

5. Summary
The reduction rate of the MHD potential energy is introduced as an energy index of tokamak plasma formationand its stability. The capacity of the stored thermal energyof the equilibrium tokamak plasma can be increased by increasing the hole current component. DCCS can results ina local minimum of the reduction rate of the MHD potential energy δWM with respect to the toroidal plasma current
distribution compared with equilibrium tokamak plasmasof the same volume. This state cannot change rapidly beyond the non-induction heating energy per time because
both increase and decrease of the toroidal plasma current
require the increase of the MHD potential energy if the coilcurrent is constant.

Acknowledgment
The author expresses his heartfelt appreciation to Prof.
Tsuguhiro Watanabe who has assisted this work with histheoretical interpretation until 2015 February. The author
is grateful to Research Institute for Applied Mechanics forgiving him the opportunity for this work.
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